| L(s) = 1 | − 0.329·2-s + 1.56·3-s − 1.89·4-s − 5-s − 0.514·6-s − 4.06·7-s + 1.28·8-s − 0.561·9-s + 0.329·10-s − 2.65·11-s − 2.95·12-s − 5.91·13-s + 1.34·14-s − 1.56·15-s + 3.35·16-s + 3.40·17-s + 0.185·18-s + 2.65·19-s + 1.89·20-s − 6.34·21-s + 0.876·22-s + 2.00·24-s + 25-s + 1.94·26-s − 5.56·27-s + 7.68·28-s + 5.84·29-s + ⋯ |
| L(s) = 1 | − 0.233·2-s + 0.901·3-s − 0.945·4-s − 0.447·5-s − 0.210·6-s − 1.53·7-s + 0.453·8-s − 0.187·9-s + 0.104·10-s − 0.801·11-s − 0.852·12-s − 1.63·13-s + 0.358·14-s − 0.403·15-s + 0.839·16-s + 0.826·17-s + 0.0436·18-s + 0.610·19-s + 0.422·20-s − 1.38·21-s + 0.186·22-s + 0.408·24-s + 0.200·25-s + 0.382·26-s − 1.07·27-s + 1.45·28-s + 1.08·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6656281506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6656281506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.329T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 + 9.31T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 0.242T + 47T^{2} \) |
| 53 | \( 1 + 9.03T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2.08T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.998906292990116158849701509742, −8.000695315639086912640183465810, −7.69260585833686400205818942743, −6.80222881040353591181891110061, −5.58190226408785063379903834263, −4.96610646152175425214644595548, −3.78256632950224626210354511769, −3.19734604892165437717720551091, −2.43389609476689129779253652490, −0.47932141218115151412402144444,
0.47932141218115151412402144444, 2.43389609476689129779253652490, 3.19734604892165437717720551091, 3.78256632950224626210354511769, 4.96610646152175425214644595548, 5.58190226408785063379903834263, 6.80222881040353591181891110061, 7.69260585833686400205818942743, 8.000695315639086912640183465810, 8.998906292990116158849701509742
